Discrete Spectrum for Nonlattice Quotients

Definitions and mathematical statement

The supplied record does not specify the ambient group $G$, the space $X$, their relationship, or the geometric and rank hypotheses. Consequently, it does not determine a unique fully quantified problem. The following formulation is conditional on the conventional analytic setting described below; these conventions require confirmation against the source context.

Suppose $G$ is a locally compact unimodular group and $\Gamma<G$ is discrete. A lattice is such a subgroup for which the invariant quotient measure satisfies $\operatorname{vol}(\Gamma\backslash G)<\infty$. Thus, under these conventions, a nonlattice satisfies $\operatorname{vol}(\Gamma\backslash G)=\infty$. Suppose also that $X$ is a Riemannian $G$-space for which the quotient $\Gamma\backslash X$, its measure, and a nonnegative self-adjoint Laplace operator $\Delta_{\Gamma\backslash X}$ on $L^2(\Gamma\backslash X)$ are defined. The intended geometry, treatment of quotient singularities, and operator domain are not supplied.

For an ambient pair $(G,X)$ in the intended, unspecified class, the first question asks whether a nonlattice $\Gamma<G$ can admit a nonzero square-integrable Laplace eigenfunction, namely whether there can exist \[ 0\ne f\in\operatorname{Dom}(\Delta_{\Gamma\backslash X}),\qquad \lambda\in[0,\infty),\qquad \Delta_{\Gamma\backslash X}f=\lambda f. \] The nonnegative eigenvalue convention here depends on the stated choice of sign for the Laplacian.

For the second question, let $\rho_\Gamma$ denote the right regular unitary representation on $L^2(\Gamma\backslash G)$, given by \[ (\rho_\Gamma(g)F)(\Gamma h)=F(\Gamma h g). \] Let $\lambda_G$ denote the left regular representation on $L^2(G)$. In the usual weak-containment sense, $\rho_\Gamma$ is tempered if $\rho_\Gamma\prec\lambda_G$. Equivalently, for every compactly supported continuous function $\varphi$ on $G$, \[ \left\|\int_G\varphi(g)\rho_\Gamma(g)\,{\,\mathrm{d}} g\right\| \le \left\|\int_G\varphi(g)\lambda_G(g)\,{\,\mathrm{d}} g\right\|. \] The second question asks whether there exists a nonlattice $\Gamma<G$ for which $\rho_\Gamma\not\prec\lambda_G$. The intended restrictions on $G$ and $\Gamma$, and whether $G$ is fixed or varies, remain unresolved. These questions concern the full displayed $L^2$ spaces; the supplied statement specifies no removal of invariant vectors or restriction to a spectral subspace.

Short English statement

For the intended group $G$ and geometric space $X$, can a discrete subgroup of infinite covolume yield a nonzero $L^2$ Laplace eigenfunction on $\Gamma\backslash X$? Can such a subgroup yield a non-tempered right regular representation on $L^2(\Gamma\backslash G)$? The ambient hypotheses are missing, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005527$, title: Discrete Spectrum for Nonlattice Quotients; exact source problem code: catalogue entry.
• Group Actions and Harmonic Analysis in Number Theory (2023), Oberwolfach Reports, pp. 1195--1262, DOI 10.4171/owr/2023/22: https://doi.org/10.4171/owr/2023/22. The supplied original excerpt identifies p. 1247. The record describes the report as documentation of a workshop and its open-problem discussions. No individual proposer is supplied.
• Supplied literature reference labeled "bottom-spectrum theorem": https://arxiv.org/abs/2304.14565.
• Supplied literature reference labeled "temperedness theorem": https://arxiv.org/abs/2202.06203.
• Literature assessment dated exactly 2026-08-22: this is supplied dated curation, not independently verified. It labels the problem partially solved and reports a negative answer to the eigenfunction existence question for nonlattices in higher-rank settings through a bottom-spectrum theorem, together with temperedness for important Anosov classes. It further reports that the full nonlattice classification and possible non-tempered cases remain open. The supplied background explicitly requires independent verification of these time-sensitive claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance assertions, not verification performed for this draft.